We evaluated the efficiency of various models in describing the time decay of aftershock rate of 47 simple sequences occurred in
California (37) from 1933 to 2004 and in Italy (10) from 1976 to 2004.We compared the models by the corrected Akaike Information
Criterion (AICc) and the Bayesian Information Criterion (BIC), both based on the log-likelihood function and also including a penalty
term that takes into account the number of independent observations and of free parameters of each model. These criteria follow two
different approaches (probabilistic and Bayesian respectively) well covering the wide spectra of current views on model comparison.
To evaluate the role of catalog incompleteness in the first times after the main shock, we compared the performance of different models
by varying the starting time Ts and the minimum magnitude threshold Mmin for each sequence. We found that Omori-type models
including parameter c are preferable to those not including it, only for short Ts and low Mmin while the latters generally perform better
than the formers for Ts longer than a few hours and Mmin larger than the main shock magnitude Mm minus 3 units. For TsN1 day or
MminNMm−2.5, only about 15% of the sequences still give a preference to models including c. This clearly indicates that a value of
parameter c different from zero does not represent a general property of aftershock sequences in California and Italy but it is very likely
induced in most cases by catalog incompleteness in the first times after the main shock.We also considered other models of aftershock
decay proposed in the literature: the Stretched Exponential Law in two forms (including and not including a time shift) and the band
Limited Power Law (LPL).We found that such models perform worse than the Modified Omori Model (MOM) and other Omori-type
models for the large majority of sequences, although for LPL, the relatively short duration of the analyzed sequences (one year) might
also contribute to its poor performance. Our analysis demonstrates that the MOMwith c kept fixed to 0 represent the better choice for
the modeling (and the forecasting) of simple sequence behavior in California and Italy.